A Monte Carlo Rendering Framework for Simulating Optical Heterodyne Detection

Abstract

Optical heterodyne detection (OHD) employs coherent light and optical interference techniques to extract physical parameters, such as velocity or distance, which are encoded in the frequency modulation of the light. With its superior signal-to-noise ratio compared to incoherent detection methods, such as time-of-flight lidar, OHD has become integral to applications requiring high sensitivity, including autonomous navigation, atmospheric sensing, and biomedical velocimetry. However, current simulation tools for OHD focus narrowly on specific applications, relying on domain-specific settings like restricted reflection functions, scene configurations, or single-bounce assumptions, which limit their applicability. In this work, we introduce a flexible and general framework for spectral-domain simulation of OHD. We demonstrate that classical radiometry-based path integral formulation can be adapted and extended to simulate the OHD measurements in the spectral domain. This enables us to leverage the rich modeling and sampling capabilities of existing Monte Carlo path tracing techniques. Our formulation shares structural similarities with transient rendering but operates in the spectral domain and accounts for the Doppler effect. While simulators for the Doppler effect in incoherent (intensity) detection methods exist, they are largely not suitable to simulate OHD. We use a microsurface interpretation to show that these two Doppler imaging techniques capture different physical quantities and thus need different simulation frameworks. We validate the correctness and predictive power of our simulation framework by qualitatively comparing the simulations with real-world captured data for three different OHD applications---FMCW lidar, blood flow velocimetry, and wind Doppler lidar.

What is OHD?

Optical heterodyne detection (OHD) is a widely used interferometric technique for measuring frequency modulations in light. OHD mixes a weak signal beam with a strong, coherent reference beam, often referred to as the local oscillator. The interference between these two optical fields generates a beat signal at a lower frequency, which encodes how the received signal is modulated relative to the reference. This often includes useful information such as the velocity or distance of objects in the scene.

A single-frequency laser is only capable of detecting a Doppler shift. In contrast, a laser with a linearly increasing frequency produces a beat frequency that contains not only the Doppler shift, but also distance (or time-of-flight) information. FMCW lidar typically uses triangular chirps, allowing us to decompose each factor.

A simple single-bounce model may work well if there is only a single object in the scene, but it fails when the scene contains complex geometry with different velocities and interreflections. This typically occurs when scanning over glossy objects or dense particles such as red blood cells or aerosols. These cases necessitate more accurate Monte Carlo simulations.

(1) Proposed OHD Path Integral

We propose a path integral formulation that considers all possible contributions to the OHD power spectrum \(S_\mathrm {AC}(\omega )\). It is important to note that we use radiometric throughput, allowing us to leverage the rich BSDF libraries in existing path tracers.

For Monte Carlo simulation, we construct \(S_\mathrm {AC}(\omega )\) by sampling each path \(\bar {\mathbf {x}}\) and adding its radiometric path throughput \(f(\bar {\mathbf {x}})\) to the corresponding frequency bin \(\omega (\bar {\mathbf {x}})\).

The path frequency term consists of two components: the optical path length \(l(\bar {\mathbf {x}})\) and the optical path velocity \(u(\bar {\mathbf {x}})\), which we have shown are related by differentiation.

(2) OHD Speckle Simulation

Since OHD uses a coherent light source, its measurements are inherently affected by speckle noise. To reproduce this effect, we propose two methods. Alg1 samples from the previously calculated power spectrum using a known distribution (negative exponential), while Alg2 adds a random phase to each sampled path and directly evaluates the optical field. Both methods depend only on the macroscopic path space.

We demonstrate the validity on a Cornell box scene, showing that both methods converge to the same PSD. However, we believe Alg1 should be the default choice, as Alg2 requires many iterations to converge. Meanwhile, the green line shows the single-bounce model, which clearly fails.

(3) Comparison with AMCW Doppler

We found that there is an underlying difference in the physics between the Doppler effect in AMCW and OHD, which results in different measurement and simulation strategies. The main reason for this difference is the large disparity in wavelength. AMCW has a much longer wavelength compared to the microgeometry; it observes the rough plane as a smooth plane and thus measures the macroscopic rate of distance change, known as spot velocity. However, for OHD, since the microgeometry has large deviations compared to the wavelength, all microscopic perturbations remain independent and we can detect the actual microscopic velocity, which is called target velocity.

BibTeX

@article{kim2025ohd, author = {Kim, Juhyeon and Benko, Craig and Wrenninge, Magnus and Villemin, Ryusuke and Barber, Zeb and Jarosz, Wojciech and Pediredla, Adithya}, title = {A Monte Carlo Rendering Framework for Simulating Optical Heterodyne Detection}, year = {2025}, issue_date = {August 2025}, publisher = {Association for Computing Machinery}, address = {New York, NY, USA}, volume = {44}, number = {4}, issn = {0730-0301}, url = {https://doi.org/10.1145/3731150}, doi = {10.1145/3731150}, journal = {ACM Trans. Graph.}, month = jul, articleno = {56}, numpages = {19} }